Two systems are said to be analogous to each other if the following two conditions are satisfied. 1. The two systems are physically different 2. Differential equation modelling of these two systems are same

1. Marathwada Mitramandal’s
COLLEGE OF ENGINEERING
Karvenagar, Pune
Accredited with ‘A’ Grade by NAAC
Presentation
On
Subject: Control System Engineering
by
Mr. A. M. Suryawanshi
Department of Electrical Engineering

2. Unit 1: Basics of Control System
Session- 5/40
Analogy between electrical and mechanical systems, block
diagram algebra
Session Plan

3. Outline:
A. Attendance
B. Review of the previous session
C. Learning Outcomes of the session
D. Content
E. Student’s evaluation
F. Preparation for next session
G. Wrap up

4. B. Review of previous session:
Problem: Modelling of Electrical and Mechanical systems (Only series linear and rotary
motion) using differential equations and transfer function

5. C. Learning Outcomes of the session:
• Develop analogy between Electrical and Mechanical systems
• Understand and apply block diagram algebra

6. D. Content:
Content
Learning /
Methodology
Faculty
Approach
Typical
Student
Activity
Skill/
Competency
Developed
Analogy between
Electrical and
Mechanical systems
PPT/Discussion Explains Understands
Critical
thinking
Block diagram
algebra
Chalk and Board
/PPT/Discussion
Explains/
Question
Understands/
Answers
Problem
solving

7. Analogy between Electrical and Mechanical
systems
Two systems are said to be analogous to each other if the following two
conditions are satisfied.
1. The two systems are physically different
2. Differential equation modelling of these two systems are same
Electrical systems and mechanical systems are two physically different systems.
There are two types of electrical analogies of translational mechanical systems.
Those are force voltage analogy and force current analogy.

8. Force Voltage Analogy
- In force voltage analogy, the mathematical equations of translational
mechanical system are compared with mesh equations of the electrical system.
𝐹 = 𝑀
𝑑2𝑥
𝑑𝑡2
+ 𝐵
𝑑𝑥
𝑑𝑡
+ 𝐾𝑥
𝑉 = 𝐿
𝑑2𝑞
𝑑𝑡2
+ 𝑅
𝑑𝑞
𝑑𝑡
+
1
𝑐
𝑞
Translational mechanical system Electrical system

9. Translational Mechanical System Electrical System
Force(F) Voltage(V)
Mass(M) Inductance(L)
Frictional Coefficient(B) Resistance(R)
Spring Constant(K) Reciprocal of Capacitance 1
𝑐
Displacement(x) Charge(q)
Velocity(v) Current(i)
𝐹 = 𝑀
𝑑2𝑥
𝑑𝑡2
+ 𝐵
𝑑𝑥
𝑑𝑡
+ 𝐾𝑥 𝑉 = 𝐿
𝑑2𝑞
𝑑𝑡2
+ 𝑅
𝑑𝑞
𝑑𝑡
+
1
𝑐
𝑞
Translational mechanical system Electrical system

10. Torque Voltage Analogy
- In this analogy, the mathematical equations of rotational mechanical
system are compared with mesh equations of the electrical system.
Rotational mechanical system
𝑉 = 𝐿
𝑑2𝑞
𝑑𝑡2
+ 𝑅
𝑑𝑞
𝑑𝑡
+
1
𝑐
𝑞
Electrical system
𝑇 = 𝐽
𝑑2
𝜃
𝑑𝑡2
+ 𝐵
𝑑𝜃
𝑑𝑡
+ 𝑘𝜃

11. Rotational Mechanical System Electrical System
Torque(T) Voltage(V)
Moment of Inertia(J) Inductance(L)
Rotational friction coefficient(B) Resistance(R)
Torsional spring constant(K) Reciprocal of Capacitance 1
𝑐
Angular Displacement(θ) Charge(q)
Angular Velocity(ω) Current(i)
Rotational mechanical system
𝑉 = 𝐿
𝑑2𝑞
𝑑𝑡2
+ 𝑅
𝑑𝑞
𝑑𝑡
+
1
𝑐
𝑞
Electrical system
𝑇 = 𝐽
𝑑2𝜃
𝑑𝑡2
+ 𝐵
𝑑𝜃
𝑑𝑡
+ 𝑘𝜃

12. Force Current Analogy
- In force current analogy, the mathematical equations of the translational
mechanical system are compared with the nodal equations of the electrical
system.
Electrical system
𝑖 =
𝑉
𝑅
+
1
𝐿
𝑉 𝑑𝑡 + 𝐶
𝑑𝑉
𝑑𝑡
𝐹 = 𝑀
𝑑2𝑥
𝑑𝑡2
+ 𝐵
𝑑𝑥
𝑑𝑡
+ 𝐾𝑥
Translational mechanical system

13. Substitute, 𝑽 =
𝒅𝝋
𝒅𝒕
in Equation
𝑖 =
𝑉
𝑅
+
1
𝐿
𝑉 𝑑𝑡 + 𝐶
𝑑𝑉
𝑑𝑡
𝑖 = 𝐶
𝑑2𝜑
𝑑𝑡2
+
1
𝑅
𝑑𝜑
𝑑𝑡
+
1
𝐿
𝜑

14. Translational Mechanical System Electrical System
Force(F) Current(i)
Mass(M) Capacitance(C)
Frictional coefficient(B) Reciprocal of Resistance 1
𝑅
Spring constant(K) Reciprocal of Inductance 1
𝐿
Displacement(x) Magnetic Flux (𝜑)
Velocity(v) Voltage(V)
Electrical system
𝐹 = 𝑀
𝑑2
𝑥
𝑑𝑡2
+ 𝐵
𝑑𝑥
𝑑𝑡
+ 𝐾𝑥
Translational mechanical system
𝑖 = 𝐶
𝑑2𝜑
𝑑𝑡2
+
1
𝑅
𝑑𝜑
𝑑𝑡
+
1
𝐿
𝜑

15. Torque Current Analogy
- In this analogy, the mathematical equations of the rotational mechanical
system are compared with the nodal mesh equations of the electrical system.
Electrical system
𝑖 =
𝑉
𝑅
+
1
𝐿
𝑉 𝑑𝑡 + 𝐶
𝑑𝑉
𝑑𝑡
Rotational mechanical system
𝑇 = 𝐽
𝑑2𝜃
𝑑𝑡2
+ 𝐵
𝑑𝜃
𝑑𝑡
+ 𝑘𝜃

16. Rotational Mechanical System Electrical System
Torque(T) Current(i)
Moment of inertia(J) Capacitance(C)
Rotational friction coefficient(B) Reciprocal of Resistance 1
𝑅
Torsional spring constant(K) Reciprocal of Inductance 1
𝐿
Angular displacement(θ) Magnetic Flux (𝜑)
Angular velocity(ω) Voltage(V)
Electrical system
Rotational mechanical system
𝑇 = 𝐽
𝑑2𝜃
𝑑𝑡2
+ 𝐵
𝑑𝜃
𝑑𝑡
+ 𝑘𝜃 𝑖 = 𝐶
𝑑2𝜑
𝑑𝑡2
+
1
𝑅
𝑑𝜑
𝑑𝑡
+
1
𝐿
𝜑

17. Block diagram algebra
- Block diagram algebra is nothing but the algebra involved with the basic
elements of the block diagram. This algebra deals with the pictorial
representation of algebraic equations.
Basic Connections for Blocks: -
There are three basic types of connections between two blocks.
1. Series Connection

18. - That means we can represent the series connection of two blocks with a single
block. The transfer function of this single block is the product of the transfer
functions of those two blocks. The equivalent block diagram is shown below.
2. Parallel Connection

19. 3. Feedback Connection
- As we discussed in previous chapters, there are two types of feedback —
positive feedback and negative feedback. The following figure shows negative
feedback control system.

20. Block Diagram Algebra for Summing Points
There are two possibilities of shifting summing points with respect to blocks −
- Shifting summing point after the block
- Shifting summing point before the block
1. Shifting Summing Point After the Block

21. 2. Shifting Summing Point Before the Block

22. Block Diagram Algebra for Take-off Points
There are two possibilities of shifting the take-off points with respect to blocks −
- Shifting take-off point after the block
- Shifting take-off point before the block
1. Shifting Take-off Point After the Block

23. 2. Shifting Take-off Point Before the Block

24. Block Diagram Reduction Rules
Follow these rules for simplifying (reducing) the block diagram, which is having many
blocks, summing points and take-off points.
Rule 1 − Check for the blocks connected in series and simplify.
Rule 2 − Check for the blocks connected in parallel and simplify.
Rule 3 − Check for the blocks connected in feedback loop and simplify.
Rule 4 − If there is difficulty with take-off point while simplifying, shift it
towards right.
Rule 5 − If there is difficulty with summing point while simplifying, shift it
towards left.
Rule 6 − Repeat the above steps till you get the simplified form, i.e., single
block.
Note − The transfer function present in this single block is the transfer function of
the overall block diagram.

25. Example
Consider the block diagram shown in the following figure. Let us simplify
(reduce) this block diagram using the block diagram reduction rules.

26. Note − Follow these steps in order to calculate the transfer function of the block
diagram having multiple inputs.
Step 1 − Find the transfer function of block diagram by considering one input at
a time and make the remaining inputs as zero.
Step 2 − Repeat step 1 for remaining inputs.
Step 3 − Get the overall transfer function by adding all those transfer
functions.

27. E. Student’s evaluation
Q1. Derive an expression for force-voltage analogy in translational and
rotational system using simple R-L-C series circuit.
Q2. Find transfer function of given system using block diagram reduction

28. Signal flow graph:
Signal flow graph is a graphical representation of algebraic equations.
Mason’s gain formula:
F. Preparation for Next Session:-
𝑇 =
𝐶(𝑠)
𝑅(𝑠)
=
𝑖=1
𝑁
𝑃𝑖∆𝑖
∆

29. G. Wrap Up
In this lecture we have learned
- Analogy between electrical and mechanical systems
Force Voltage Analogy
Torque Voltage Analogy
Force Current Analogy
Torque Current Analogy
- Block diagram algebra
Basic Connections for Blocks
Block Diagram Algebra for Summing Points
Block Diagram Algebra for Take-off Points